/*
 * Prime generation.
 */

#include <assert.h>
#include "ssh.h"

/*
 * This prime generation algorithm is pretty much cribbed from
 * OpenSSL. The algorithm is:
 *
 *  - invent a B-bit random number and ensure the top and bottom
 *    bits are set (so it's definitely B-bit, and it's definitely
 *    odd)
 *
 *  - see if it's coprime to all primes below 2^16; increment it by
 *    two until it is (this shouldn't take long in general)
 *
 *  - perform the Miller-Rabin primality test enough times to
 *    ensure the probability of it being composite is 2^-80 or
 *    less
 *
 *  - go back to square one if any M-R test fails.
 */

/*
 * The Miller-Rabin primality test is an extension to the Fermat
 * test. The Fermat test just checks that a^(p-1) == 1 mod p; this
 * is vulnerable to Carmichael numbers. Miller-Rabin considers how
 * that 1 is derived as well.
 *
 * Lemma: if a^2 == 1 (mod p), and p is prime, then either a == 1
 * or a == -1 (mod p).
 *
 *   Proof: p divides a^2-1, i.e. p divides (a+1)(a-1). Hence,
 *   since p is prime, either p divides (a+1) or p divides (a-1).
 *   But this is the same as saying that either a is congruent to
 *   -1 mod p or a is congruent to +1 mod p. []
 *
 *   Comment: This fails when p is not prime. Consider p=mn, so
 *   that mn divides (a+1)(a-1). Now we could have m dividing (a+1)
 *   and n dividing (a-1), without the whole of mn dividing either.
 *   For example, consider a=10 and p=99. 99 = 9 * 11; 9 divides
 *   10-1 and 11 divides 10+1, so a^2 is congruent to 1 mod p
 *   without a having to be congruent to either 1 or -1.
 *
 * So the Miller-Rabin test, as well as considering a^(p-1),
 * considers a^((p-1)/2), a^((p-1)/4), and so on as far as it can
 * go. In other words. we write p-1 as q * 2^k, with k as large as
 * possible (i.e. q must be odd), and we consider the powers
 *
 *       a^(q*2^0)      a^(q*2^1)          ...  a^(q*2^(k-1))  a^(q*2^k)
 * i.e.  a^((n-1)/2^k)  a^((n-1)/2^(k-1))  ...  a^((n-1)/2)    a^(n-1)
 *
 * If p is to be prime, the last of these must be 1. Therefore, by
 * the above lemma, the one before it must be either 1 or -1. And
 * _if_ it's 1, then the one before that must be either 1 or -1,
 * and so on ... In other words, we expect to see a trailing chain
 * of 1s preceded by a -1. (If we're unlucky, our trailing chain of
 * 1s will be as long as the list so we'll never get to see what
 * lies before it. This doesn't count as a test failure because it
 * hasn't _proved_ that p is not prime.)
 *
 * For example, consider a=2 and p=1729. 1729 is a Carmichael
 * number: although it's not prime, it satisfies a^(p-1) == 1 mod p
 * for any a coprime to it. So the Fermat test wouldn't have a
 * problem with it at all, unless we happened to stumble on an a
 * which had a common factor.
 *
 * So. 1729 - 1 equals 27 * 2^6. So we look at
 *
 *     2^27 mod 1729 == 645
 *    2^108 mod 1729 == 1065
 *    2^216 mod 1729 == 1
 *    2^432 mod 1729 == 1
 *    2^864 mod 1729 == 1
 *   2^1728 mod 1729 == 1
 *
 * We do have a trailing string of 1s, so the Fermat test would
 * have been happy. But this trailing string of 1s is preceded by
 * 1065; whereas if 1729 were prime, we'd expect to see it preceded
 * by -1 (i.e. 1728.). Guards! Seize this impostor.
 *
 * (If we were unlucky, we might have tried a=16 instead of a=2;
 * now 16^27 mod 1729 == 1, so we would have seen a long string of
 * 1s and wouldn't have seen the thing _before_ the 1s. So, just
 * like the Fermat test, for a given p there may well exist values
 * of a which fail to show up its compositeness. So we try several,
 * just like the Fermat test. The difference is that Miller-Rabin
 * is not _in general_ fooled by Carmichael numbers.)
 *
 * Put simply, then, the Miller-Rabin test requires us to:
 *
 *  1. write p-1 as q * 2^k, with q odd
 *  2. compute z = (a^q) mod p.
 *  3. report success if z == 1 or z == -1.
 *  4. square z at most k-1 times, and report success if it becomes
 *     -1 at any point.
 *  5. report failure otherwise.
 *
 * (We expect z to become -1 after at most k-1 squarings, because
 * if it became -1 after k squarings then a^(p-1) would fail to be
 * 1. And we don't need to investigate what happens after we see a
 * -1, because we _know_ that -1 squared is 1 modulo anything at
 * all, so after we've seen a -1 we can be sure of seeing nothing
 * but 1s.)
 */

/*
 * The first few odd primes.
 *
 * import sys
 * def sieve(n):
 *     z = []
 *     list = []
 *     for i in range(n): z.append(1)
 *     for i in range(2,n):
 *         if z[i]:
 *             list.append(i)
 *             for j in range(i,n,i): z[j] = 0
 *     return list
 * list = sieve(65535)
 * for i in list[1:]: sys.stdout.write("%d," % i)
 */
static const unsigned short primes[] = {
    3,     5,     7,     11,    13,    17,    19,    23,    29,    31,    37,
    41,    43,    47,    53,    59,    61,    67,    71,    73,    79,    83,
    89,    97,    101,   103,   107,   109,   113,   127,   131,   137,   139,
    149,   151,   157,   163,   167,   173,   179,   181,   191,   193,   197,
    199,   211,   223,   227,   229,   233,   239,   241,   251,   257,   263,
    269,   271,   277,   281,   283,   293,   307,   311,   313,   317,   331,
    337,   347,   349,   353,   359,   367,   373,   379,   383,   389,   397,
    401,   409,   419,   421,   431,   433,   439,   443,   449,   457,   461,
    463,   467,   479,   487,   491,   499,   503,   509,   521,   523,   541,
    547,   557,   563,   569,   571,   577,   587,   593,   599,   601,   607,
    613,   617,   619,   631,   641,   643,   647,   653,   659,   661,   673,
    677,   683,   691,   701,   709,   719,   727,   733,   739,   743,   751,
    757,   761,   769,   773,   787,   797,   809,   811,   821,   823,   827,
    829,   839,   853,   857,   859,   863,   877,   881,   883,   887,   907,
    911,   919,   929,   937,   941,   947,   953,   967,   971,   977,   983,
    991,   997,   1009,  1013,  1019,  1021,  1031,  1033,  1039,  1049,  1051,
    1061,  1063,  1069,  1087,  1091,  1093,  1097,  1103,  1109,  1117,  1123,
    1129,  1151,  1153,  1163,  1171,  1181,  1187,  1193,  1201,  1213,  1217,
    1223,  1229,  1231,  1237,  1249,  1259,  1277,  1279,  1283,  1289,  1291,
    1297,  1301,  1303,  1307,  1319,  1321,  1327,  1361,  1367,  1373,  1381,
    1399,  1409,  1423,  1427,  1429,  1433,  1439,  1447,  1451,  1453,  1459,
    1471,  1481,  1483,  1487,  1489,  1493,  1499,  1511,  1523,  1531,  1543,
    1549,  1553,  1559,  1567,  1571,  1579,  1583,  1597,  1601,  1607,  1609,
    1613,  1619,  1621,  1627,  1637,  1657,  1663,  1667,  1669,  1693,  1697,
    1699,  1709,  1721,  1723,  1733,  1741,  1747,  1753,  1759,  1777,  1783,
    1787,  1789,  1801,  1811,  1823,  1831,  1847,  1861,  1867,  1871,  1873,
    1877,  1879,  1889,  1901,  1907,  1913,  1931,  1933,  1949,  1951,  1973,
    1979,  1987,  1993,  1997,  1999,  2003,  2011,  2017,  2027,  2029,  2039,
    2053,  2063,  2069,  2081,  2083,  2087,  2089,  2099,  2111,  2113,  2129,
    2131,  2137,  2141,  2143,  2153,  2161,  2179,  2203,  2207,  2213,  2221,
    2237,  2239,  2243,  2251,  2267,  2269,  2273,  2281,  2287,  2293,  2297,
    2309,  2311,  2333,  2339,  2341,  2347,  2351,  2357,  2371,  2377,  2381,
    2383,  2389,  2393,  2399,  2411,  2417,  2423,  2437,  2441,  2447,  2459,
    2467,  2473,  2477,  2503,  2521,  2531,  2539,  2543,  2549,  2551,  2557,
    2579,  2591,  2593,  2609,  2617,  2621,  2633,  2647,  2657,  2659,  2663,
    2671,  2677,  2683,  2687,  2689,  2693,  2699,  2707,  2711,  2713,  2719,
    2729,  2731,  2741,  2749,  2753,  2767,  2777,  2789,  2791,  2797,  2801,
    2803,  2819,  2833,  2837,  2843,  2851,  2857,  2861,  2879,  2887,  2897,
    2903,  2909,  2917,  2927,  2939,  2953,  2957,  2963,  2969,  2971,  2999,
    3001,  3011,  3019,  3023,  3037,  3041,  3049,  3061,  3067,  3079,  3083,
    3089,  3109,  3119,  3121,  3137,  3163,  3167,  3169,  3181,  3187,  3191,
    3203,  3209,  3217,  3221,  3229,  3251,  3253,  3257,  3259,  3271,  3299,
    3301,  3307,  3313,  3319,  3323,  3329,  3331,  3343,  3347,  3359,  3361,
    3371,  3373,  3389,  3391,  3407,  3413,  3433,  3449,  3457,  3461,  3463,
    3467,  3469,  3491,  3499,  3511,  3517,  3527,  3529,  3533,  3539,  3541,
    3547,  3557,  3559,  3571,  3581,  3583,  3593,  3607,  3613,  3617,  3623,
    3631,  3637,  3643,  3659,  3671,  3673,  3677,  3691,  3697,  3701,  3709,
    3719,  3727,  3733,  3739,  3761,  3767,  3769,  3779,  3793,  3797,  3803,
    3821,  3823,  3833,  3847,  3851,  3853,  3863,  3877,  3881,  3889,  3907,
    3911,  3917,  3919,  3923,  3929,  3931,  3943,  3947,  3967,  3989,  4001,
    4003,  4007,  4013,  4019,  4021,  4027,  4049,  4051,  4057,  4073,  4079,
    4091,  4093,  4099,  4111,  4127,  4129,  4133,  4139,  4153,  4157,  4159,
    4177,  4201,  4211,  4217,  4219,  4229,  4231,  4241,  4243,  4253,  4259,
    4261,  4271,  4273,  4283,  4289,  4297,  4327,  4337,  4339,  4349,  4357,
    4363,  4373,  4391,  4397,  4409,  4421,  4423,  4441,  4447,  4451,  4457,
    4463,  4481,  4483,  4493,  4507,  4513,  4517,  4519,  4523,  4547,  4549,
    4561,  4567,  4583,  4591,  4597,  4603,  4621,  4637,  4639,  4643,  4649,
    4651,  4657,  4663,  4673,  4679,  4691,  4703,  4721,  4723,  4729,  4733,
    4751,  4759,  4783,  4787,  4789,  4793,  4799,  4801,  4813,  4817,  4831,
    4861,  4871,  4877,  4889,  4903,  4909,  4919,  4931,  4933,  4937,  4943,
    4951,  4957,  4967,  4969,  4973,  4987,  4993,  4999,  5003,  5009,  5011,
    5021,  5023,  5039,  5051,  5059,  5077,  5081,  5087,  5099,  5101,  5107,
    5113,  5119,  5147,  5153,  5167,  5171,  5179,  5189,  5197,  5209,  5227,
    5231,  5233,  5237,  5261,  5273,  5279,  5281,  5297,  5303,  5309,  5323,
    5333,  5347,  5351,  5381,  5387,  5393,  5399,  5407,  5413,  5417,  5419,
    5431,  5437,  5441,  5443,  5449,  5471,  5477,  5479,  5483,  5501,  5503,
    5507,  5519,  5521,  5527,  5531,  5557,  5563,  5569,  5573,  5581,  5591,
    5623,  5639,  5641,  5647,  5651,  5653,  5657,  5659,  5669,  5683,  5689,
    5693,  5701,  5711,  5717,  5737,  5741,  5743,  5749,  5779,  5783,  5791,
    5801,  5807,  5813,  5821,  5827,  5839,  5843,  5849,  5851,  5857,  5861,
    5867,  5869,  5879,  5881,  5897,  5903,  5923,  5927,  5939,  5953,  5981,
    5987,  6007,  6011,  6029,  6037,  6043,  6047,  6053,  6067,  6073,  6079,
    6089,  6091,  6101,  6113,  6121,  6131,  6133,  6143,  6151,  6163,  6173,
    6197,  6199,  6203,  6211,  6217,  6221,  6229,  6247,  6257,  6263,  6269,
    6271,  6277,  6287,  6299,  6301,  6311,  6317,  6323,  6329,  6337,  6343,
    6353,  6359,  6361,  6367,  6373,  6379,  6389,  6397,  6421,  6427,  6449,
    6451,  6469,  6473,  6481,  6491,  6521,  6529,  6547,  6551,  6553,  6563,
    6569,  6571,  6577,  6581,  6599,  6607,  6619,  6637,  6653,  6659,  6661,
    6673,  6679,  6689,  6691,  6701,  6703,  6709,  6719,  6733,  6737,  6761,
    6763,  6779,  6781,  6791,  6793,  6803,  6823,  6827,  6829,  6833,  6841,
    6857,  6863,  6869,  6871,  6883,  6899,  6907,  6911,  6917,  6947,  6949,
    6959,  6961,  6967,  6971,  6977,  6983,  6991,  6997,  7001,  7013,  7019,
    7027,  7039,  7043,  7057,  7069,  7079,  7103,  7109,  7121,  7127,  7129,
    7151,  7159,  7177,  7187,  7193,  7207,  7211,  7213,  7219,  7229,  7237,
    7243,  7247,  7253,  7283,  7297,  7307,  7309,  7321,  7331,  7333,  7349,
    7351,  7369,  7393,  7411,  7417,  7433,  7451,  7457,  7459,  7477,  7481,
    7487,  7489,  7499,  7507,  7517,  7523,  7529,  7537,  7541,  7547,  7549,
    7559,  7561,  7573,  7577,  7583,  7589,  7591,  7603,  7607,  7621,  7639,
    7643,  7649,  7669,  7673,  7681,  7687,  7691,  7699,  7703,  7717,  7723,
    7727,  7741,  7753,  7757,  7759,  7789,  7793,  7817,  7823,  7829,  7841,
    7853,  7867,  7873,  7877,  7879,  7883,  7901,  7907,  7919,  7927,  7933,
    7937,  7949,  7951,  7963,  7993,  8009,  8011,  8017,  8039,  8053,  8059,
    8069,  8081,  8087,  8089,  8093,  8101,  8111,  8117,  8123,  8147,  8161,
    8167,  8171,  8179,  8191,  8209,  8219,  8221,  8231,  8233,  8237,  8243,
    8263,  8269,  8273,  8287,  8291,  8293,  8297,  8311,  8317,  8329,  8353,
    8363,  8369,  8377,  8387,  8389,  8419,  8423,  8429,  8431,  8443,  8447,
    8461,  8467,  8501,  8513,  8521,  8527,  8537,  8539,  8543,  8563,  8573,
    8581,  8597,  8599,  8609,  8623,  8627,  8629,  8641,  8647,  8663,  8669,
    8677,  8681,  8689,  8693,  8699,  8707,  8713,  8719,  8731,  8737,  8741,
    8747,  8753,  8761,  8779,  8783,  8803,  8807,  8819,  8821,  8831,  8837,
    8839,  8849,  8861,  8863,  8867,  8887,  8893,  8923,  8929,  8933,  8941,
    8951,  8963,  8969,  8971,  8999,  9001,  9007,  9011,  9013,  9029,  9041,
    9043,  9049,  9059,  9067,  9091,  9103,  9109,  9127,  9133,  9137,  9151,
    9157,  9161,  9173,  9181,  9187,  9199,  9203,  9209,  9221,  9227,  9239,
    9241,  9257,  9277,  9281,  9283,  9293,  9311,  9319,  9323,  9337,  9341,
    9343,  9349,  9371,  9377,  9391,  9397,  9403,  9413,  9419,  9421,  9431,
    9433,  9437,  9439,  9461,  9463,  9467,  9473,  9479,  9491,  9497,  9511,
    9521,  9533,  9539,  9547,  9551,  9587,  9601,  9613,  9619,  9623,  9629,
    9631,  9643,  9649,  9661,  9677,  9679,  9689,  9697,  9719,  9721,  9733,
    9739,  9743,  9749,  9767,  9769,  9781,  9787,  9791,  9803,  9811,  9817,
    9829,  9833,  9839,  9851,  9857,  9859,  9871,  9883,  9887,  9901,  9907,
    9923,  9929,  9931,  9941,  9949,  9967,  9973,  10007, 10009, 10037, 10039,
    10061, 10067, 10069, 10079, 10091, 10093, 10099, 10103, 10111, 10133, 10139,
    10141, 10151, 10159, 10163, 10169, 10177, 10181, 10193, 10211, 10223, 10243,
    10247, 10253, 10259, 10267, 10271, 10273, 10289, 10301, 10303, 10313, 10321,
    10331, 10333, 10337, 10343, 10357, 10369, 10391, 10399, 10427, 10429, 10433,
    10453, 10457, 10459, 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531,
    10559, 10567, 10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651,
    10657, 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739,
    10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, 10861,
    10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, 10957, 10973,
    10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, 11069, 11071, 11083,
    11087, 11093, 11113, 11117, 11119, 11131, 11149, 11159, 11161, 11171, 11173,
    11177, 11197, 11213, 11239, 11243, 11251, 11257, 11261, 11273, 11279, 11287,
    11299, 11311, 11317, 11321, 11329, 11351, 11353, 11369, 11383, 11393, 11399,
    11411, 11423, 11437, 11443, 11447, 11467, 11471, 11483, 11489, 11491, 11497,
    11503, 11519, 11527, 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621,
    11633, 11657, 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743,
    11777, 11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833,
    11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, 11939,
    11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011, 12037, 12041,
    12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109, 12113, 12119, 12143,
    12149, 12157, 12161, 12163, 12197, 12203, 12211, 12227, 12239, 12241, 12251,
    12253, 12263, 12269, 12277, 12281, 12289, 12301, 12323, 12329, 12343, 12347,
    12373, 12377, 12379, 12391, 12401, 12409, 12413, 12421, 12433, 12437, 12451,
    12457, 12473, 12479, 12487, 12491, 12497, 12503, 12511, 12517, 12527, 12539,
    12541, 12547, 12553, 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619,
    12637, 12641, 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721,
    12739, 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829,
    12841, 12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, 12941,
    12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007, 13009, 13033,
    13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109, 13121, 13127, 13147,
    13151, 13159, 13163, 13171, 13177, 13183, 13187, 13217, 13219, 13229, 13241,
    13249, 13259, 13267, 13291, 13297, 13309, 13313, 13327, 13331, 13337, 13339,
    13367, 13381, 13397, 13399, 13411, 13417, 13421, 13441, 13451, 13457, 13463,
    13469, 13477, 13487, 13499, 13513, 13523, 13537, 13553, 13567, 13577, 13591,
    13597, 13613, 13619, 13627, 13633, 13649, 13669, 13679, 13681, 13687, 13691,
    13693, 13697, 13709, 13711, 13721, 13723, 13729, 13751, 13757, 13759, 13763,
    13781, 13789, 13799, 13807, 13829, 13831, 13841, 13859, 13873, 13877, 13879,
    13883, 13901, 13903, 13907, 13913, 13921, 13931, 13933, 13963, 13967, 13997,
    13999, 14009, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087,
    14107, 14143, 14149, 14153, 14159, 14173, 14177, 14197, 14207, 14221, 14243,
    14249, 14251, 14281, 14293, 14303, 14321, 14323, 14327, 14341, 14347, 14369,
    14387, 14389, 14401, 14407, 14411, 14419, 14423, 14431, 14437, 14447, 14449,
    14461, 14479, 14489, 14503, 14519, 14533, 14537, 14543, 14549, 14551, 14557,
    14561, 14563, 14591, 14593, 14621, 14627, 14629, 14633, 14639, 14653, 14657,
    14669, 14683, 14699, 14713, 14717, 14723, 14731, 14737, 14741, 14747, 14753,
    14759, 14767, 14771, 14779, 14783, 14797, 14813, 14821, 14827, 14831, 14843,
    14851, 14867, 14869, 14879, 14887, 14891, 14897, 14923, 14929, 14939, 14947,
    14951, 14957, 14969, 14983, 15013, 15017, 15031, 15053, 15061, 15073, 15077,
    15083, 15091, 15101, 15107, 15121, 15131, 15137, 15139, 15149, 15161, 15173,
    15187, 15193, 15199, 15217, 15227, 15233, 15241, 15259, 15263, 15269, 15271,
    15277, 15287, 15289, 15299, 15307, 15313, 15319, 15329, 15331, 15349, 15359,
    15361, 15373, 15377, 15383, 15391, 15401, 15413, 15427, 15439, 15443, 15451,
    15461, 15467, 15473, 15493, 15497, 15511, 15527, 15541, 15551, 15559, 15569,
    15581, 15583, 15601, 15607, 15619, 15629, 15641, 15643, 15647, 15649, 15661,
    15667, 15671, 15679, 15683, 15727, 15731, 15733, 15737, 15739, 15749, 15761,
    15767, 15773, 15787, 15791, 15797, 15803, 15809, 15817, 15823, 15859, 15877,
    15881, 15887, 15889, 15901, 15907, 15913, 15919, 15923, 15937, 15959, 15971,
    15973, 15991, 16001, 16007, 16033, 16057, 16061, 16063, 16067, 16069, 16073,
    16087, 16091, 16097, 16103, 16111, 16127, 16139, 16141, 16183, 16187, 16189,
    16193, 16217, 16223, 16229, 16231, 16249, 16253, 16267, 16273, 16301, 16319,
    16333, 16339, 16349, 16361, 16363, 16369, 16381, 16411, 16417, 16421, 16427,
    16433, 16447, 16451, 16453, 16477, 16481, 16487, 16493, 16519, 16529, 16547,
    16553, 16561, 16567, 16573, 16603, 16607, 16619, 16631, 16633, 16649, 16651,
    16657, 16661, 16673, 16691, 16693, 16699, 16703, 16729, 16741, 16747, 16759,
    16763, 16787, 16811, 16823, 16829, 16831, 16843, 16871, 16879, 16883, 16889,
    16901, 16903, 16921, 16927, 16931, 16937, 16943, 16963, 16979, 16981, 16987,
    16993, 17011, 17021, 17027, 17029, 17033, 17041, 17047, 17053, 17077, 17093,
    17099, 17107, 17117, 17123, 17137, 17159, 17167, 17183, 17189, 17191, 17203,
    17207, 17209, 17231, 17239, 17257, 17291, 17293, 17299, 17317, 17321, 17327,
    17333, 17341, 17351, 17359, 17377, 17383, 17387, 17389, 17393, 17401, 17417,
    17419, 17431, 17443, 17449, 17467, 17471, 17477, 17483, 17489, 17491, 17497,
    17509, 17519, 17539, 17551, 17569, 17573, 17579, 17581, 17597, 17599, 17609,
    17623, 17627, 17657, 17659, 17669, 17681, 17683, 17707, 17713, 17729, 17737,
    17747, 17749, 17761, 17783, 17789, 17791, 17807, 17827, 17837, 17839, 17851,
    17863, 17881, 17891, 17903, 17909, 17911, 17921, 17923, 17929, 17939, 17957,
    17959, 17971, 17977, 17981, 17987, 17989, 18013, 18041, 18043, 18047, 18049,
    18059, 18061, 18077, 18089, 18097, 18119, 18121, 18127, 18131, 18133, 18143,
    18149, 18169, 18181, 18191, 18199, 18211, 18217, 18223, 18229, 18233, 18251,
    18253, 18257, 18269, 18287, 18289, 18301, 18307, 18311, 18313, 18329, 18341,
    18353, 18367, 18371, 18379, 18397, 18401, 18413, 18427, 18433, 18439, 18443,
    18451, 18457, 18461, 18481, 18493, 18503, 18517, 18521, 18523, 18539, 18541,
    18553, 18583, 18587, 18593, 18617, 18637, 18661, 18671, 18679, 18691, 18701,
    18713, 18719, 18731, 18743, 18749, 18757, 18773, 18787, 18793, 18797, 18803,
    18839, 18859, 18869, 18899, 18911, 18913, 18917, 18919, 18947, 18959, 18973,
    18979, 19001, 19009, 19013, 19031, 19037, 19051, 19069, 19073, 19079, 19081,
    19087, 19121, 19139, 19141, 19157, 19163, 19181, 19183, 19207, 19211, 19213,
    19219, 19231, 19237, 19249, 19259, 19267, 19273, 19289, 19301, 19309, 19319,
    19333, 19373, 19379, 19381, 19387, 19391, 19403, 19417, 19421, 19423, 19427,
    19429, 19433, 19441, 19447, 19457, 19463, 19469, 19471, 19477, 19483, 19489,
    19501, 19507, 19531, 19541, 19543, 19553, 19559, 19571, 19577, 19583, 19597,
    19603, 19609, 19661, 19681, 19687, 19697, 19699, 19709, 19717, 19727, 19739,
    19751, 19753, 19759, 19763, 19777, 19793, 19801, 19813, 19819, 19841, 19843,
    19853, 19861, 19867, 19889, 19891, 19913, 19919, 19927, 19937, 19949, 19961,
    19963, 19973, 19979, 19991, 19993, 19997, 20011, 20021, 20023, 20029, 20047,
    20051, 20063, 20071, 20089, 20101, 20107, 20113, 20117, 20123, 20129, 20143,
    20147, 20149, 20161, 20173, 20177, 20183, 20201, 20219, 20231, 20233, 20249,
    20261, 20269, 20287, 20297, 20323, 20327, 20333, 20341, 20347, 20353, 20357,
    20359, 20369, 20389, 20393, 20399, 20407, 20411, 20431, 20441, 20443, 20477,
    20479, 20483, 20507, 20509, 20521, 20533, 20543, 20549, 20551, 20563, 20593,
    20599, 20611, 20627, 20639, 20641, 20663, 20681, 20693, 20707, 20717, 20719,
    20731, 20743, 20747, 20749, 20753, 20759, 20771, 20773, 20789, 20807, 20809,
    20849, 20857, 20873, 20879, 20887, 20897, 20899, 20903, 20921, 20929, 20939,
    20947, 20959, 20963, 20981, 20983, 21001, 21011, 21013, 21017, 21019, 21023,
    21031, 21059, 21061, 21067, 21089, 21101, 21107, 21121, 21139, 21143, 21149,
    21157, 21163, 21169, 21179, 21187, 21191, 21193, 21211, 21221, 21227, 21247,
    21269, 21277, 21283, 21313, 21317, 21319, 21323, 21341, 21347, 21377, 21379,
    21383, 21391, 21397, 21401, 21407, 21419, 21433, 21467, 21481, 21487, 21491,
    21493, 21499, 21503, 21517, 21521, 21523, 21529, 21557, 21559, 21563, 21569,
    21577, 21587, 21589, 21599, 21601, 21611, 21613, 21617, 21647, 21649, 21661,
    21673, 21683, 21701, 21713, 21727, 21737, 21739, 21751, 21757, 21767, 21773,
    21787, 21799, 21803, 21817, 21821, 21839, 21841, 21851, 21859, 21863, 21871,
    21881, 21893, 21911, 21929, 21937, 21943, 21961, 21977, 21991, 21997, 22003,
    22013, 22027, 22031, 22037, 22039, 22051, 22063, 22067, 22073, 22079, 22091,
    22093, 22109, 22111, 22123, 22129, 22133, 22147, 22153, 22157, 22159, 22171,
    22189, 22193, 22229, 22247, 22259, 22271, 22273, 22277, 22279, 22283, 22291,
    22303, 22307, 22343, 22349, 22367, 22369, 22381, 22391, 22397, 22409, 22433,
    22441, 22447, 22453, 22469, 22481, 22483, 22501, 22511, 22531, 22541, 22543,
    22549, 22567, 22571, 22573, 22613, 22619, 22621, 22637, 22639, 22643, 22651,
    22669, 22679, 22691, 22697, 22699, 22709, 22717, 22721, 22727, 22739, 22741,
    22751, 22769, 22777, 22783, 22787, 22807, 22811, 22817, 22853, 22859, 22861,
    22871, 22877, 22901, 22907, 22921, 22937, 22943, 22961, 22963, 22973, 22993,
    23003, 23011, 23017, 23021, 23027, 23029, 23039, 23041, 23053, 23057, 23059,
    23063, 23071, 23081, 23087, 23099, 23117, 23131, 23143, 23159, 23167, 23173,
    23189, 23197, 23201, 23203, 23209, 23227, 23251, 23269, 23279, 23291, 23293,
    23297, 23311, 23321, 23327, 23333, 23339, 23357, 23369, 23371, 23399, 23417,
    23431, 23447, 23459, 23473, 23497, 23509, 23531, 23537, 23539, 23549, 23557,
    23561, 23563, 23567, 23581, 23593, 23599, 23603, 23609, 23623, 23627, 23629,
    23633, 23663, 23669, 23671, 23677, 23687, 23689, 23719, 23741, 23743, 23747,
    23753, 23761, 23767, 23773, 23789, 23801, 23813, 23819, 23827, 23831, 23833,
    23857, 23869, 23873, 23879, 23887, 23893, 23899, 23909, 23911, 23917, 23929,
    23957, 23971, 23977, 23981, 23993, 24001, 24007, 24019, 24023, 24029, 24043,
    24049, 24061, 24071, 24077, 24083, 24091, 24097, 24103, 24107, 24109, 24113,
    24121, 24133, 24137, 24151, 24169, 24179, 24181, 24197, 24203, 24223, 24229,
    24239, 24247, 24251, 24281, 24317, 24329, 24337, 24359, 24371, 24373, 24379,
    24391, 24407, 24413, 24419, 24421, 24439, 24443, 24469, 24473, 24481, 24499,
    24509, 24517, 24527, 24533, 24547, 24551, 24571, 24593, 24611, 24623, 24631,
    24659, 24671, 24677, 24683, 24691, 24697, 24709, 24733, 24749, 24763, 24767,
    24781, 24793, 24799, 24809, 24821, 24841, 24847, 24851, 24859, 24877, 24889,
    24907, 24917, 24919, 24923, 24943, 24953, 24967, 24971, 24977, 24979, 24989,
    25013, 25031, 25033, 25037, 25057, 25073, 25087, 25097, 25111, 25117, 25121,
    25127, 25147, 25153, 25163, 25169, 25171, 25183, 25189, 25219, 25229, 25237,
    25243, 25247, 25253, 25261, 25301, 25303, 25307, 25309, 25321, 25339, 25343,
    25349, 25357, 25367, 25373, 25391, 25409, 25411, 25423, 25439, 25447, 25453,
    25457, 25463, 25469, 25471, 25523, 25537, 25541, 25561, 25577, 25579, 25583,
    25589, 25601, 25603, 25609, 25621, 25633, 25639, 25643, 25657, 25667, 25673,
    25679, 25693, 25703, 25717, 25733, 25741, 25747, 25759, 25763, 25771, 25793,
    25799, 25801, 25819, 25841, 25847, 25849, 25867, 25873, 25889, 25903, 25913,
    25919, 25931, 25933, 25939, 25943, 25951, 25969, 25981, 25997, 25999, 26003,
    26017, 26021, 26029, 26041, 26053, 26083, 26099, 26107, 26111, 26113, 26119,
    26141, 26153, 26161, 26171, 26177, 26183, 26189, 26203, 26209, 26227, 26237,
    26249, 26251, 26261, 26263, 26267, 26293, 26297, 26309, 26317, 26321, 26339,
    26347, 26357, 26371, 26387, 26393, 26399, 26407, 26417, 26423, 26431, 26437,
    26449, 26459, 26479, 26489, 26497, 26501, 26513, 26539, 26557, 26561, 26573,
    26591, 26597, 26627, 26633, 26641, 26647, 26669, 26681, 26683, 26687, 26693,
    26699, 26701, 26711, 26713, 26717, 26723, 26729, 26731, 26737, 26759, 26777,
    26783, 26801, 26813, 26821, 26833, 26839, 26849, 26861, 26863, 26879, 26881,
    26891, 26893, 26903, 26921, 26927, 26947, 26951, 26953, 26959, 26981, 26987,
    26993, 27011, 27017, 27031, 27043, 27059, 27061, 27067, 27073, 27077, 27091,
    27103, 27107, 27109, 27127, 27143, 27179, 27191, 27197, 27211, 27239, 27241,
    27253, 27259, 27271, 27277, 27281, 27283, 27299, 27329, 27337, 27361, 27367,
    27397, 27407, 27409, 27427, 27431, 27437, 27449, 27457, 27479, 27481, 27487,
    27509, 27527, 27529, 27539, 27541, 27551, 27581, 27583, 27611, 27617, 27631,
    27647, 27653, 27673, 27689, 27691, 27697, 27701, 27733, 27737, 27739, 27743,
    27749, 27751, 27763, 27767, 27773, 27779, 27791, 27793, 27799, 27803, 27809,
    27817, 27823, 27827, 27847, 27851, 27883, 27893, 27901, 27917, 27919, 27941,
    27943, 27947, 27953, 27961, 27967, 27983, 27997, 28001, 28019, 28027, 28031,
    28051, 28057, 28069, 28081, 28087, 28097, 28099, 28109, 28111, 28123, 28151,
    28163, 28181, 28183, 28201, 28211, 28219, 28229, 28277, 28279, 28283, 28289,
    28297, 28307, 28309, 28319, 28349, 28351, 28387, 28393, 28403, 28409, 28411,
    28429, 28433, 28439, 28447, 28463, 28477, 28493, 28499, 28513, 28517, 28537,
    28541, 28547, 28549, 28559, 28571, 28573, 28579, 28591, 28597, 28603, 28607,
    28619, 28621, 28627, 28631, 28643, 28649, 28657, 28661, 28663, 28669, 28687,
    28697, 28703, 28711, 28723, 28729, 28751, 28753, 28759, 28771, 28789, 28793,
    28807, 28813, 28817, 28837, 28843, 28859, 28867, 28871, 28879, 28901, 28909,
    28921, 28927, 28933, 28949, 28961, 28979, 29009, 29017, 29021, 29023, 29027,
    29033, 29059, 29063, 29077, 29101, 29123, 29129, 29131, 29137, 29147, 29153,
    29167, 29173, 29179, 29191, 29201, 29207, 29209, 29221, 29231, 29243, 29251,
    29269, 29287, 29297, 29303, 29311, 29327, 29333, 29339, 29347, 29363, 29383,
    29387, 29389, 29399, 29401, 29411, 29423, 29429, 29437, 29443, 29453, 29473,
    29483, 29501, 29527, 29531, 29537, 29567, 29569, 29573, 29581, 29587, 29599,
    29611, 29629, 29633, 29641, 29663, 29669, 29671, 29683, 29717, 29723, 29741,
    29753, 29759, 29761, 29789, 29803, 29819, 29833, 29837, 29851, 29863, 29867,
    29873, 29879, 29881, 29917, 29921, 29927, 29947, 29959, 29983, 29989, 30011,
    30013, 30029, 30047, 30059, 30071, 30089, 30091, 30097, 30103, 30109, 30113,
    30119, 30133, 30137, 30139, 30161, 30169, 30181, 30187, 30197, 30203, 30211,
    30223, 30241, 30253, 30259, 30269, 30271, 30293, 30307, 30313, 30319, 30323,
    30341, 30347, 30367, 30389, 30391, 30403, 30427, 30431, 30449, 30467, 30469,
    30491, 30493, 30497, 30509, 30517, 30529, 30539, 30553, 30557, 30559, 30577,
    30593, 30631, 30637, 30643, 30649, 30661, 30671, 30677, 30689, 30697, 30703,
    30707, 30713, 30727, 30757, 30763, 30773, 30781, 30803, 30809, 30817, 30829,
    30839, 30841, 30851, 30853, 30859, 30869, 30871, 30881, 30893, 30911, 30931,
    30937, 30941, 30949, 30971, 30977, 30983, 31013, 31019, 31033, 31039, 31051,
    31063, 31069, 31079, 31081, 31091, 31121, 31123, 31139, 31147, 31151, 31153,
    31159, 31177, 31181, 31183, 31189, 31193, 31219, 31223, 31231, 31237, 31247,
    31249, 31253, 31259, 31267, 31271, 31277, 31307, 31319, 31321, 31327, 31333,
    31337, 31357, 31379, 31387, 31391, 31393, 31397, 31469, 31477, 31481, 31489,
    31511, 31513, 31517, 31531, 31541, 31543, 31547, 31567, 31573, 31583, 31601,
    31607, 31627, 31643, 31649, 31657, 31663, 31667, 31687, 31699, 31721, 31723,
    31727, 31729, 31741, 31751, 31769, 31771, 31793, 31799, 31817, 31847, 31849,
    31859, 31873, 31883, 31891, 31907, 31957, 31963, 31973, 31981, 31991, 32003,
    32009, 32027, 32029, 32051, 32057, 32059, 32063, 32069, 32077, 32083, 32089,
    32099, 32117, 32119, 32141, 32143, 32159, 32173, 32183, 32189, 32191, 32203,
    32213, 32233, 32237, 32251, 32257, 32261, 32297, 32299, 32303, 32309, 32321,
    32323, 32327, 32341, 32353, 32359, 32363, 32369, 32371, 32377, 32381, 32401,
    32411, 32413, 32423, 32429, 32441, 32443, 32467, 32479, 32491, 32497, 32503,
    32507, 32531, 32533, 32537, 32561, 32563, 32569, 32573, 32579, 32587, 32603,
    32609, 32611, 32621, 32633, 32647, 32653, 32687, 32693, 32707, 32713, 32717,
    32719, 32749, 32771, 32779, 32783, 32789, 32797, 32801, 32803, 32831, 32833,
    32839, 32843, 32869, 32887, 32909, 32911, 32917, 32933, 32939, 32941, 32957,
    32969, 32971, 32983, 32987, 32993, 32999, 33013, 33023, 33029, 33037, 33049,
    33053, 33071, 33073, 33083, 33091, 33107, 33113, 33119, 33149, 33151, 33161,
    33179, 33181, 33191, 33199, 33203, 33211, 33223, 33247, 33287, 33289, 33301,
    33311, 33317, 33329, 33331, 33343, 33347, 33349, 33353, 33359, 33377, 33391,
    33403, 33409, 33413, 33427, 33457, 33461, 33469, 33479, 33487, 33493, 33503,
    33521, 33529, 33533, 33547, 33563, 33569, 33577, 33581, 33587, 33589, 33599,
    33601, 33613, 33617, 33619, 33623, 33629, 33637, 33641, 33647, 33679, 33703,
    33713, 33721, 33739, 33749, 33751, 33757, 33767, 33769, 33773, 33791, 33797,
    33809, 33811, 33827, 33829, 33851, 33857, 33863, 33871, 33889, 33893, 33911,
    33923, 33931, 33937, 33941, 33961, 33967, 33997, 34019, 34031, 34033, 34039,
    34057, 34061, 34123, 34127, 34129, 34141, 34147, 34157, 34159, 34171, 34183,
    34211, 34213, 34217, 34231, 34253, 34259, 34261, 34267, 34273, 34283, 34297,
    34301, 34303, 34313, 34319, 34327, 34337, 34351, 34361, 34367, 34369, 34381,
    34403, 34421, 34429, 34439, 34457, 34469, 34471, 34483, 34487, 34499, 34501,
    34511, 34513, 34519, 34537, 34543, 34549, 34583, 34589, 34591, 34603, 34607,
    34613, 34631, 34649, 34651, 34667, 34673, 34679, 34687, 34693, 34703, 34721,
    34729, 34739, 34747, 34757, 34759, 34763, 34781, 34807, 34819, 34841, 34843,
    34847, 34849, 34871, 34877, 34883, 34897, 34913, 34919, 34939, 34949, 34961,
    34963, 34981, 35023, 35027, 35051, 35053, 35059, 35069, 35081, 35083, 35089,
    35099, 35107, 35111, 35117, 35129, 35141, 35149, 35153, 35159, 35171, 35201,
    35221, 35227, 35251, 35257, 35267, 35279, 35281, 35291, 35311, 35317, 35323,
    35327, 35339, 35353, 35363, 35381, 35393, 35401, 35407, 35419, 35423, 35437,
    35447, 35449, 35461, 35491, 35507, 35509, 35521, 35527, 35531, 35533, 35537,
    35543, 35569, 35573, 35591, 35593, 35597, 35603, 35617, 35671, 35677, 35729,
    35731, 35747, 35753, 35759, 35771, 35797, 35801, 35803, 35809, 35831, 35837,
    35839, 35851, 35863, 35869, 35879, 35897, 35899, 35911, 35923, 35933, 35951,
    35963, 35969, 35977, 35983, 35993, 35999, 36007, 36011, 36013, 36017, 36037,
    36061, 36067, 36073, 36083, 36097, 36107, 36109, 36131, 36137, 36151, 36161,
    36187, 36191, 36209, 36217, 36229, 36241, 36251, 36263, 36269, 36277, 36293,
    36299, 36307, 36313, 36319, 36341, 36343, 36353, 36373, 36383, 36389, 36433,
    36451, 36457, 36467, 36469, 36473, 36479, 36493, 36497, 36523, 36527, 36529,
    36541, 36551, 36559, 36563, 36571, 36583, 36587, 36599, 36607, 36629, 36637,
    36643, 36653, 36671, 36677, 36683, 36691, 36697, 36709, 36713, 36721, 36739,
    36749, 36761, 36767, 36779, 36781, 36787, 36791, 36793, 36809, 36821, 36833,
    36847, 36857, 36871, 36877, 36887, 36899, 36901, 36913, 36919, 36923, 36929,
    36931, 36943, 36947, 36973, 36979, 36997, 37003, 37013, 37019, 37021, 37039,
    37049, 37057, 37061, 37087, 37097, 37117, 37123, 37139, 37159, 37171, 37181,
    37189, 37199, 37201, 37217, 37223, 37243, 37253, 37273, 37277, 37307, 37309,
    37313, 37321, 37337, 37339, 37357, 37361, 37363, 37369, 37379, 37397, 37409,
    37423, 37441, 37447, 37463, 37483, 37489, 37493, 37501, 37507, 37511, 37517,
    37529, 37537, 37547, 37549, 37561, 37567, 37571, 37573, 37579, 37589, 37591,
    37607, 37619, 37633, 37643, 37649, 37657, 37663, 37691, 37693, 37699, 37717,
    37747, 37781, 37783, 37799, 37811, 37813, 37831, 37847, 37853, 37861, 37871,
    37879, 37889, 37897, 37907, 37951, 37957, 37963, 37967, 37987, 37991, 37993,
    37997, 38011, 38039, 38047, 38053, 38069, 38083, 38113, 38119, 38149, 38153,
    38167, 38177, 38183, 38189, 38197, 38201, 38219, 38231, 38237, 38239, 38261,
    38273, 38281, 38287, 38299, 38303, 38317, 38321, 38327, 38329, 38333, 38351,
    38371, 38377, 38393, 38431, 38447, 38449, 38453, 38459, 38461, 38501, 38543,
    38557, 38561, 38567, 38569, 38593, 38603, 38609, 38611, 38629, 38639, 38651,
    38653, 38669, 38671, 38677, 38693, 38699, 38707, 38711, 38713, 38723, 38729,
    38737, 38747, 38749, 38767, 38783, 38791, 38803, 38821, 38833, 38839, 38851,
    38861, 38867, 38873, 38891, 38903, 38917, 38921, 38923, 38933, 38953, 38959,
    38971, 38977, 38993, 39019, 39023, 39041, 39043, 39047, 39079, 39089, 39097,
    39103, 39107, 39113, 39119, 39133, 39139, 39157, 39161, 39163, 39181, 39191,
    39199, 39209, 39217, 39227, 39229, 39233, 39239, 39241, 39251, 39293, 39301,
    39313, 39317, 39323, 39341, 39343, 39359, 39367, 39371, 39373, 39383, 39397,
    39409, 39419, 39439, 39443, 39451, 39461, 39499, 39503, 39509, 39511, 39521,
    39541, 39551, 39563, 39569, 39581, 39607, 39619, 39623, 39631, 39659, 39667,
    39671, 39679, 39703, 39709, 39719, 39727, 39733, 39749, 39761, 39769, 39779,
    39791, 39799, 39821, 39827, 39829, 39839, 39841, 39847, 39857, 39863, 39869,
    39877, 39883, 39887, 39901, 39929, 39937, 39953, 39971, 39979, 39983, 39989,
    40009, 40013, 40031, 40037, 40039, 40063, 40087, 40093, 40099, 40111, 40123,
    40127, 40129, 40151, 40153, 40163, 40169, 40177, 40189, 40193, 40213, 40231,
    40237, 40241, 40253, 40277, 40283, 40289, 40343, 40351, 40357, 40361, 40387,
    40423, 40427, 40429, 40433, 40459, 40471, 40483, 40487, 40493, 40499, 40507,
    40519, 40529, 40531, 40543, 40559, 40577, 40583, 40591, 40597, 40609, 40627,
    40637, 40639, 40693, 40697, 40699, 40709, 40739, 40751, 40759, 40763, 40771,
    40787, 40801, 40813, 40819, 40823, 40829, 40841, 40847, 40849, 40853, 40867,
    40879, 40883, 40897, 40903, 40927, 40933, 40939, 40949, 40961, 40973, 40993,
    41011, 41017, 41023, 41039, 41047, 41051, 41057, 41077, 41081, 41113, 41117,
    41131, 41141, 41143, 41149, 41161, 41177, 41179, 41183, 41189, 41201, 41203,
    41213, 41221, 41227, 41231, 41233, 41243, 41257, 41263, 41269, 41281, 41299,
    41333, 41341, 41351, 41357, 41381, 41387, 41389, 41399, 41411, 41413, 41443,
    41453, 41467, 41479, 41491, 41507, 41513, 41519, 41521, 41539, 41543, 41549,
    41579, 41593, 41597, 41603, 41609, 41611, 41617, 41621, 41627, 41641, 41647,
    41651, 41659, 41669, 41681, 41687, 41719, 41729, 41737, 41759, 41761, 41771,
    41777, 41801, 41809, 41813, 41843, 41849, 41851, 41863, 41879, 41887, 41893,
    41897, 41903, 41911, 41927, 41941, 41947, 41953, 41957, 41959, 41969, 41981,
    41983, 41999, 42013, 42017, 42019, 42023, 42043, 42061, 42071, 42073, 42083,
    42089, 42101, 42131, 42139, 42157, 42169, 42179, 42181, 42187, 42193, 42197,
    42209, 42221, 42223, 42227, 42239, 42257, 42281, 42283, 42293, 42299, 42307,
    42323, 42331, 42337, 42349, 42359, 42373, 42379, 42391, 42397, 42403, 42407,
    42409, 42433, 42437, 42443, 42451, 42457, 42461, 42463, 42467, 42473, 42487,
    42491, 42499, 42509, 42533, 42557, 42569, 42571, 42577, 42589, 42611, 42641,
    42643, 42649, 42667, 42677, 42683, 42689, 42697, 42701, 42703, 42709, 42719,
    42727, 42737, 42743, 42751, 42767, 42773, 42787, 42793, 42797, 42821, 42829,
    42839, 42841, 42853, 42859, 42863, 42899, 42901, 42923, 42929, 42937, 42943,
    42953, 42961, 42967, 42979, 42989, 43003, 43013, 43019, 43037, 43049, 43051,
    43063, 43067, 43093, 43103, 43117, 43133, 43151, 43159, 43177, 43189, 43201,
    43207, 43223, 43237, 43261, 43271, 43283, 43291, 43313, 43319, 43321, 43331,
    43391, 43397, 43399, 43403, 43411, 43427, 43441, 43451, 43457, 43481, 43487,
    43499, 43517, 43541, 43543, 43573, 43577, 43579, 43591, 43597, 43607, 43609,
    43613, 43627, 43633, 43649, 43651, 43661, 43669, 43691, 43711, 43717, 43721,
    43753, 43759, 43777, 43781, 43783, 43787, 43789, 43793, 43801, 43853, 43867,
    43889, 43891, 43913, 43933, 43943, 43951, 43961, 43963, 43969, 43973, 43987,
    43991, 43997, 44017, 44021, 44027, 44029, 44041, 44053, 44059, 44071, 44087,
    44089, 44101, 44111, 44119, 44123, 44129, 44131, 44159, 44171, 44179, 44189,
    44201, 44203, 44207, 44221, 44249, 44257, 44263, 44267, 44269, 44273, 44279,
    44281, 44293, 44351, 44357, 44371, 44381, 44383, 44389, 44417, 44449, 44453,
    44483, 44491, 44497, 44501, 44507, 44519, 44531, 44533, 44537, 44543, 44549,
    44563, 44579, 44587, 44617, 44621, 44623, 44633, 44641, 44647, 44651, 44657,
    44683, 44687, 44699, 44701, 44711, 44729, 44741, 44753, 44771, 44773, 44777,
    44789, 44797, 44809, 44819, 44839, 44843, 44851, 44867, 44879, 44887, 44893,
    44909, 44917, 44927, 44939, 44953, 44959, 44963, 44971, 44983, 44987, 45007,
    45013, 45053, 45061, 45077, 45083, 45119, 45121, 45127, 45131, 45137, 45139,
    45161, 45179, 45181, 45191, 45197, 45233, 45247, 45259, 45263, 45281, 45289,
    45293, 45307, 45317, 45319, 45329, 45337, 45341, 45343, 45361, 45377, 45389,
    45403, 45413, 45427, 45433, 45439, 45481, 45491, 45497, 45503, 45523, 45533,
    45541, 45553, 45557, 45569, 45587, 45589, 45599, 45613, 45631, 45641, 45659,
    45667, 45673, 45677, 45691, 45697, 45707, 45737, 45751, 45757, 45763, 45767,
    45779, 45817, 45821, 45823, 45827, 45833, 45841, 45853, 45863, 45869, 45887,
    45893, 45943, 45949, 45953, 45959, 45971, 45979, 45989, 46021, 46027, 46049,
    46051, 46061, 46073, 46091, 46093, 46099, 46103, 46133, 46141, 46147, 46153,
    46171, 46181, 46183, 46187, 46199, 46219, 46229, 46237, 46261, 46271, 46273,
    46279, 46301, 46307, 46309, 46327, 46337, 46349, 46351, 46381, 46399, 46411,
    46439, 46441, 46447, 46451, 46457, 46471, 46477, 46489, 46499, 46507, 46511,
    46523, 46549, 46559, 46567, 46573, 46589, 46591, 46601, 46619, 46633, 46639,
    46643, 46649, 46663, 46679, 46681, 46687, 46691, 46703, 46723, 46727, 46747,
    46751, 46757, 46769, 46771, 46807, 46811, 46817, 46819, 46829, 46831, 46853,
    46861, 46867, 46877, 46889, 46901, 46919, 46933, 46957, 46993, 46997, 47017,
    47041, 47051, 47057, 47059, 47087, 47093, 47111, 47119, 47123, 47129, 47137,
    47143, 47147, 47149, 47161, 47189, 47207, 47221, 47237, 47251, 47269, 47279,
    47287, 47293, 47297, 47303, 47309, 47317, 47339, 47351, 47353, 47363, 47381,
    47387, 47389, 47407, 47417, 47419, 47431, 47441, 47459, 47491, 47497, 47501,
    47507, 47513, 47521, 47527, 47533, 47543, 47563, 47569, 47581, 47591, 47599,
    47609, 47623, 47629, 47639, 47653, 47657, 47659, 47681, 47699, 47701, 47711,
    47713, 47717, 47737, 47741, 47743, 47777, 47779, 47791, 47797, 47807, 47809,
    47819, 47837, 47843, 47857, 47869, 47881, 47903, 47911, 47917, 47933, 47939,
    47947, 47951, 47963, 47969, 47977, 47981, 48017, 48023, 48029, 48049, 48073,
    48079, 48091, 48109, 48119, 48121, 48131, 48157, 48163, 48179, 48187, 48193,
    48197, 48221, 48239, 48247, 48259, 48271, 48281, 48299, 48311, 48313, 48337,
    48341, 48353, 48371, 48383, 48397, 48407, 48409, 48413, 48437, 48449, 48463,
    48473, 48479, 48481, 48487, 48491, 48497, 48523, 48527, 48533, 48539, 48541,
    48563, 48571, 48589, 48593, 48611, 48619, 48623, 48647, 48649, 48661, 48673,
    48677, 48679, 48731, 48733, 48751, 48757, 48761, 48767, 48779, 48781, 48787,
    48799, 48809, 48817, 48821, 48823, 48847, 48857, 48859, 48869, 48871, 48883,
    48889, 48907, 48947, 48953, 48973, 48989, 48991, 49003, 49009, 49019, 49031,
    49033, 49037, 49043, 49057, 49069, 49081, 49103, 49109, 49117, 49121, 49123,
    49139, 49157, 49169, 49171, 49177, 49193, 49199, 49201, 49207, 49211, 49223,
    49253, 49261, 49277, 49279, 49297, 49307, 49331, 49333, 49339, 49363, 49367,
    49369, 49391, 49393, 49409, 49411, 49417, 49429, 49433, 49451, 49459, 49463,
    49477, 49481, 49499, 49523, 49529, 49531, 49537, 49547, 49549, 49559, 49597,
    49603, 49613, 49627, 49633, 49639, 49663, 49667, 49669, 49681, 49697, 49711,
    49727, 49739, 49741, 49747, 49757, 49783, 49787, 49789, 49801, 49807, 49811,
    49823, 49831, 49843, 49853, 49871, 49877, 49891, 49919, 49921, 49927, 49937,
    49939, 49943, 49957, 49991, 49993, 49999, 50021, 50023, 50033, 50047, 50051,
    50053, 50069, 50077, 50087, 50093, 50101, 50111, 50119, 50123, 50129, 50131,
    50147, 50153, 50159, 50177, 50207, 50221, 50227, 50231, 50261, 50263, 50273,
    50287, 50291, 50311, 50321, 50329, 50333, 50341, 50359, 50363, 50377, 50383,
    50387, 50411, 50417, 50423, 50441, 50459, 50461, 50497, 50503, 50513, 50527,
    50539, 50543, 50549, 50551, 50581, 50587, 50591, 50593, 50599, 50627, 50647,
    50651, 50671, 50683, 50707, 50723, 50741, 50753, 50767, 50773, 50777, 50789,
    50821, 50833, 50839, 50849, 50857, 50867, 50873, 50891, 50893, 50909, 50923,
    50929, 50951, 50957, 50969, 50971, 50989, 50993, 51001, 51031, 51043, 51047,
    51059, 51061, 51071, 51109, 51131, 51133, 51137, 51151, 51157, 51169, 51193,
    51197, 51199, 51203, 51217, 51229, 51239, 51241, 51257, 51263, 51283, 51287,
    51307, 51329, 51341, 51343, 51347, 51349, 51361, 51383, 51407, 51413, 51419,
    51421, 51427, 51431, 51437, 51439, 51449, 51461, 51473, 51479, 51481, 51487,
    51503, 51511, 51517, 51521, 51539, 51551, 51563, 51577, 51581, 51593, 51599,
    51607, 51613, 51631, 51637, 51647, 51659, 51673, 51679, 51683, 51691, 51713,
    51719, 51721, 51749, 51767, 51769, 51787, 51797, 51803, 51817, 51827, 51829,
    51839, 51853, 51859, 51869, 51871, 51893, 51899, 51907, 51913, 51929, 51941,
    51949, 51971, 51973, 51977, 51991, 52009, 52021, 52027, 52051, 52057, 52067,
    52069, 52081, 52103, 52121, 52127, 52147, 52153, 52163, 52177, 52181, 52183,
    52189, 52201, 52223, 52237, 52249, 52253, 52259, 52267, 52289, 52291, 52301,
    52313, 52321, 52361, 52363, 52369, 52379, 52387, 52391, 52433, 52453, 52457,
    52489, 52501, 52511, 52517, 52529, 52541, 52543, 52553, 52561, 52567, 52571,
    52579, 52583, 52609, 52627, 52631, 52639, 52667, 52673, 52691, 52697, 52709,
    52711, 52721, 52727, 52733, 52747, 52757, 52769, 52783, 52807, 52813, 52817,
    52837, 52859, 52861, 52879, 52883, 52889, 52901, 52903, 52919, 52937, 52951,
    52957, 52963, 52967, 52973, 52981, 52999, 53003, 53017, 53047, 53051, 53069,
    53077, 53087, 53089, 53093, 53101, 53113, 53117, 53129, 53147, 53149, 53161,
    53171, 53173, 53189, 53197, 53201, 53231, 53233, 53239, 53267, 53269, 53279,
    53281, 53299, 53309, 53323, 53327, 53353, 53359, 53377, 53381, 53401, 53407,
    53411, 53419, 53437, 53441, 53453, 53479, 53503, 53507, 53527, 53549, 53551,
    53569, 53591, 53593, 53597, 53609, 53611, 53617, 53623, 53629, 53633, 53639,
    53653, 53657, 53681, 53693, 53699, 53717, 53719, 53731, 53759, 53773, 53777,
    53783, 53791, 53813, 53819, 53831, 53849, 53857, 53861, 53881, 53887, 53891,
    53897, 53899, 53917, 53923, 53927, 53939, 53951, 53959, 53987, 53993, 54001,
    54011, 54013, 54037, 54049, 54059, 54083, 54091, 54101, 54121, 54133, 54139,
    54151, 54163, 54167, 54181, 54193, 54217, 54251, 54269, 54277, 54287, 54293,
    54311, 54319, 54323, 54331, 54347, 54361, 54367, 54371, 54377, 54401, 54403,
    54409, 54413, 54419, 54421, 54437, 54443, 54449, 54469, 54493, 54497, 54499,
    54503, 54517, 54521, 54539, 54541, 54547, 54559, 54563, 54577, 54581, 54583,
    54601, 54617, 54623, 54629, 54631, 54647, 54667, 54673, 54679, 54709, 54713,
    54721, 54727, 54751, 54767, 54773, 54779, 54787, 54799, 54829, 54833, 54851,
    54869, 54877, 54881, 54907, 54917, 54919, 54941, 54949, 54959, 54973, 54979,
    54983, 55001, 55009, 55021, 55049, 55051, 55057, 55061, 55073, 55079, 55103,
    55109, 55117, 55127, 55147, 55163, 55171, 55201, 55207, 55213, 55217, 55219,
    55229, 55243, 55249, 55259, 55291, 55313, 55331, 55333, 55337, 55339, 55343,
    55351, 55373, 55381, 55399, 55411, 55439, 55441, 55457, 55469, 55487, 55501,
    55511, 55529, 55541, 55547, 55579, 55589, 55603, 55609, 55619, 55621, 55631,
    55633, 55639, 55661, 55663, 55667, 55673, 55681, 55691, 55697, 55711, 55717,
    55721, 55733, 55763, 55787, 55793, 55799, 55807, 55813, 55817, 55819, 55823,
    55829, 55837, 55843, 55849, 55871, 55889, 55897, 55901, 55903, 55921, 55927,
    55931, 55933, 55949, 55967, 55987, 55997, 56003, 56009, 56039, 56041, 56053,
    56081, 56087, 56093, 56099, 56101, 56113, 56123, 56131, 56149, 56167, 56171,
    56179, 56197, 56207, 56209, 56237, 56239, 56249, 56263, 56267, 56269, 56299,
    56311, 56333, 56359, 56369, 56377, 56383, 56393, 56401, 56417, 56431, 56437,
    56443, 56453, 56467, 56473, 56477, 56479, 56489, 56501, 56503, 56509, 56519,
    56527, 56531, 56533, 56543, 56569, 56591, 56597, 56599, 56611, 56629, 56633,
    56659, 56663, 56671, 56681, 56687, 56701, 56711, 56713, 56731, 56737, 56747,
    56767, 56773, 56779, 56783, 56807, 56809, 56813, 56821, 56827, 56843, 56857,
    56873, 56891, 56893, 56897, 56909, 56911, 56921, 56923, 56929, 56941, 56951,
    56957, 56963, 56983, 56989, 56993, 56999, 57037, 57041, 57047, 57059, 57073,
    57077, 57089, 57097, 57107, 57119, 57131, 57139, 57143, 57149, 57163, 57173,
    57179, 57191, 57193, 57203, 57221, 57223, 57241, 57251, 57259, 57269, 57271,
    57283, 57287, 57301, 57329, 57331, 57347, 57349, 57367, 57373, 57383, 57389,
    57397, 57413, 57427, 57457, 57467, 57487, 57493, 57503, 57527, 57529, 57557,
    57559, 57571, 57587, 57593, 57601, 57637, 57641, 57649, 57653, 57667, 57679,
    57689, 57697, 57709, 57713, 57719, 57727, 57731, 57737, 57751, 57773, 57781,
    57787, 57791, 57793, 57803, 57809, 57829, 57839, 57847, 57853, 57859, 57881,
    57899, 57901, 57917, 57923, 57943, 57947, 57973, 57977, 57991, 58013, 58027,
    58031, 58043, 58049, 58057, 58061, 58067, 58073, 58099, 58109, 58111, 58129,
    58147, 58151, 58153, 58169, 58171, 58189, 58193, 58199, 58207, 58211, 58217,
    58229, 58231, 58237, 58243, 58271, 58309, 58313, 58321, 58337, 58363, 58367,
    58369, 58379, 58391, 58393, 58403, 58411, 58417, 58427, 58439, 58441, 58451,
    58453, 58477, 58481, 58511, 58537, 58543, 58549, 58567, 58573, 58579, 58601,
    58603, 58613, 58631, 58657, 58661, 58679, 58687, 58693, 58699, 58711, 58727,
    58733, 58741, 58757, 58763, 58771, 58787, 58789, 58831, 58889, 58897, 58901,
    58907, 58909, 58913, 58921, 58937, 58943, 58963, 58967, 58979, 58991, 58997,
    59009, 59011, 59021, 59023, 59029, 59051, 59053, 59063, 59069, 59077, 59083,
    59093, 59107, 59113, 59119, 59123, 59141, 59149, 59159, 59167, 59183, 59197,
    59207, 59209, 59219, 59221, 59233, 59239, 59243, 59263, 59273, 59281, 59333,
    59341, 59351, 59357, 59359, 59369, 59377, 59387, 59393, 59399, 59407, 59417,
    59419, 59441, 59443, 59447, 59453, 59467, 59471, 59473, 59497, 59509, 59513,
    59539, 59557, 59561, 59567, 59581, 59611, 59617, 59621, 59627, 59629, 59651,
    59659, 59663, 59669, 59671, 59693, 59699, 59707, 59723, 59729, 59743, 59747,
    59753, 59771, 59779, 59791, 59797, 59809, 59833, 59863, 59879, 59887, 59921,
    59929, 59951, 59957, 59971, 59981, 59999, 60013, 60017, 60029, 60037, 60041,
    60077, 60083, 60089, 60091, 60101, 60103, 60107, 60127, 60133, 60139, 60149,
    60161, 60167, 60169, 60209, 60217, 60223, 60251, 60257, 60259, 60271, 60289,
    60293, 60317, 60331, 60337, 60343, 60353, 60373, 60383, 60397, 60413, 60427,
    60443, 60449, 60457, 60493, 60497, 60509, 60521, 60527, 60539, 60589, 60601,
    60607, 60611, 60617, 60623, 60631, 60637, 60647, 60649, 60659, 60661, 60679,
    60689, 60703, 60719, 60727, 60733, 60737, 60757, 60761, 60763, 60773, 60779,
    60793, 60811, 60821, 60859, 60869, 60887, 60889, 60899, 60901, 60913, 60917,
    60919, 60923, 60937, 60943, 60953, 60961, 61001, 61007, 61027, 61031, 61043,
    61051, 61057, 61091, 61099, 61121, 61129, 61141, 61151, 61153, 61169, 61211,
    61223, 61231, 61253, 61261, 61283, 61291, 61297, 61331, 61333, 61339, 61343,
    61357, 61363, 61379, 61381, 61403, 61409, 61417, 61441, 61463, 61469, 61471,
    61483, 61487, 61493, 61507, 61511, 61519, 61543, 61547, 61553, 61559, 61561,
    61583, 61603, 61609, 61613, 61627, 61631, 61637, 61643, 61651, 61657, 61667,
    61673, 61681, 61687, 61703, 61717, 61723, 61729, 61751, 61757, 61781, 61813,
    61819, 61837, 61843, 61861, 61871, 61879, 61909, 61927, 61933, 61949, 61961,
    61967, 61979, 61981, 61987, 61991, 62003, 62011, 62017, 62039, 62047, 62053,
    62057, 62071, 62081, 62099, 62119, 62129, 62131, 62137, 62141, 62143, 62171,
    62189, 62191, 62201, 62207, 62213, 62219, 62233, 62273, 62297, 62299, 62303,
    62311, 62323, 62327, 62347, 62351, 62383, 62401, 62417, 62423, 62459, 62467,
    62473, 62477, 62483, 62497, 62501, 62507, 62533, 62539, 62549, 62563, 62581,
    62591, 62597, 62603, 62617, 62627, 62633, 62639, 62653, 62659, 62683, 62687,
    62701, 62723, 62731, 62743, 62753, 62761, 62773, 62791, 62801, 62819, 62827,
    62851, 62861, 62869, 62873, 62897, 62903, 62921, 62927, 62929, 62939, 62969,
    62971, 62981, 62983, 62987, 62989, 63029, 63031, 63059, 63067, 63073, 63079,
    63097, 63103, 63113, 63127, 63131, 63149, 63179, 63197, 63199, 63211, 63241,
    63247, 63277, 63281, 63299, 63311, 63313, 63317, 63331, 63337, 63347, 63353,
    63361, 63367, 63377, 63389, 63391, 63397, 63409, 63419, 63421, 63439, 63443,
    63463, 63467, 63473, 63487, 63493, 63499, 63521, 63527, 63533, 63541, 63559,
    63577, 63587, 63589, 63599, 63601, 63607, 63611, 63617, 63629, 63647, 63649,
    63659, 63667, 63671, 63689, 63691, 63697, 63703, 63709, 63719, 63727, 63737,
    63743, 63761, 63773, 63781, 63793, 63799, 63803, 63809, 63823, 63839, 63841,
    63853, 63857, 63863, 63901, 63907, 63913, 63929, 63949, 63977, 63997, 64007,
    64013, 64019, 64033, 64037, 64063, 64067, 64081, 64091, 64109, 64123, 64151,
    64153, 64157, 64171, 64187, 64189, 64217, 64223, 64231, 64237, 64271, 64279,
    64283, 64301, 64303, 64319, 64327, 64333, 64373, 64381, 64399, 64403, 64433,
    64439, 64451, 64453, 64483, 64489, 64499, 64513, 64553, 64567, 64577, 64579,
    64591, 64601, 64609, 64613, 64621, 64627, 64633, 64661, 64663, 64667, 64679,
    64693, 64709, 64717, 64747, 64763, 64781, 64783, 64793, 64811, 64817, 64849,
    64853, 64871, 64877, 64879, 64891, 64901, 64919, 64921, 64927, 64937, 64951,
    64969, 64997, 65003, 65011, 65027, 65029, 65033, 65053, 65063, 65071, 65089,
    65099, 65101, 65111, 65119, 65123, 65129, 65141, 65147, 65167, 65171, 65173,
    65179, 65183, 65203, 65213, 65239, 65257, 65267, 65269, 65287, 65293, 65309,
    65323, 65327, 65353, 65357, 65371, 65381, 65393, 65407, 65413, 65419, 65423,
    65437, 65447, 65449, 65479, 65497, 65519, 65521,
};

#define NPRIMES (sizeof(primes) / sizeof(*primes))

/*
 * Generate a prime. We can deal with various extra properties of
 * the prime:
 *
 *  - to speed up use in RSA, we can arrange to select a prime with
 *    the property (prime % modulus) != residue.
 *
 *  - for use in DSA, we can arrange to select a prime which is one
 *    more than a multiple of a dirty great bignum. In this case
 *    `bits' gives the size of the factor by which we _multiply_
 *    that bignum, rather than the size of the whole number.
 *
 *  - for the basically cosmetic purposes of generating keys of the
 *    length actually specified rather than off by one bit, we permit
 *    the caller to provide an unsigned integer 'firstbits' which will
 *    match the top few bits of the returned prime. (That is, there
 *    will exist some n such that (returnvalue >> n) == firstbits.) If
 *    'firstbits' is not needed, specifying it to either 0 or 1 is
 *    an adequate no-op.
 */
Bignum primegen(int bits,
                int modulus,
                int residue,
                Bignum factor,
                int phase,
                progfn_t pfn,
                void *pfnparam,
                unsigned firstbits)
{
  int i, k, v, byte, bitsleft, check, checks, fbsize;
  unsigned long delta;
  unsigned long moduli[NPRIMES + 1];
  unsigned long residues[NPRIMES + 1];
  unsigned long multipliers[NPRIMES + 1];
  Bignum p, pm1, q, wqp, wqp2;
  int progress = 0;

  byte = 0;
  bitsleft = 0;

  fbsize = 0;
  while (firstbits >> fbsize) /* work out how to align this */
    fbsize++;

STARTOVER:

  pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);

  /*
   * Generate a k-bit random number with top and bottom bits set.
   * Alternatively, if `factor' is nonzero, generate a k-bit
   * random number with the top bit set and the bottom bit clear,
   * multiply it by `factor', and add one.
   */
  p = bn_power_2(bits - 1);
  for (i = 0; i < bits; i++) {
    if (i == 0 || i == bits - 1) {
      v = (i != 0 || !factor) ? 1 : 0;
    } else if (i >= bits - fbsize) {
      v = (firstbits >> (i - (bits - fbsize))) & 1;
    } else {
      if (bitsleft <= 0)
        bitsleft = 8, byte = random_byte();
      v = byte & 1;
      byte >>= 1;
      bitsleft--;
    }
    bignum_set_bit(p, i, v);
  }
  if (factor) {
    Bignum tmp = p;
    p = bigmul(tmp, factor);
    freebn(tmp);
    assert(bignum_bit(p, 0) == 0);
    bignum_set_bit(p, 0, 1);
  }

  /*
   * Ensure this random number is coprime to the first few
   * primes, by repeatedly adding either 2 or 2*factor to it
   * until it is.
   */
  for (i = 0; i < NPRIMES; i++) {
    moduli[i] = primes[i];
    residues[i] = bignum_mod_short(p, primes[i]);
    if (factor)
      multipliers[i] = bignum_mod_short(factor, primes[i]);
    else
      multipliers[i] = 1;
  }
  moduli[NPRIMES] = modulus;
  residues[NPRIMES] =
      (bignum_mod_short(p, (unsigned short)modulus) + modulus - residue);
  if (factor)
    multipliers[NPRIMES] = bignum_mod_short(factor, modulus);
  else
    multipliers[NPRIMES] = 1;
  delta = 0;
  while (1) {
    for (i = 0; i < (sizeof(moduli) / sizeof(*moduli)); i++)
      if (!((residues[i] + delta * multipliers[i]) % moduli[i]))
        break;
    if (i < (sizeof(moduli) / sizeof(*moduli))) { /* we broke */
      delta += 2;
      if (delta > 65536) {
        freebn(p);
        goto STARTOVER;
      }
      continue;
    }
    break;
  }
  q = p;
  if (factor) {
    Bignum tmp;
    tmp = bignum_from_long(delta);
    p = bigmuladd(tmp, factor, q);
    freebn(tmp);
  } else {
    p = bignum_add_long(q, delta);
  }
  freebn(q);

  /*
   * Now apply the Miller-Rabin primality test a few times. First
   * work out how many checks are needed.
   */
  checks = 27;
  if (bits >= 150)
    checks = 18;
  if (bits >= 200)
    checks = 15;
  if (bits >= 250)
    checks = 12;
  if (bits >= 300)
    checks = 9;
  if (bits >= 350)
    checks = 8;
  if (bits >= 400)
    checks = 7;
  if (bits >= 450)
    checks = 6;
  if (bits >= 550)
    checks = 5;
  if (bits >= 650)
    checks = 4;
  if (bits >= 850)
    checks = 3;
  if (bits >= 1300)
    checks = 2;

  /*
   * Next, write p-1 as q*2^k.
   */
  for (k = 0; bignum_bit(p, k) == !k; k++)
    continue; /* find first 1 bit in p-1 */
  q = bignum_rshift(p, k);
  /* And store p-1 itself, which we'll need. */
  pm1 = copybn(p);
  decbn(pm1);

  /*
   * Now, for each check ...
   */
  for (check = 0; check < checks; check++) {
    Bignum w;

    /*
     * Invent a random number between 1 and p-1 inclusive.
     */
    while (1) {
      w = bn_power_2(bits - 1);
      for (i = 0; i < bits; i++) {
        if (bitsleft <= 0)
          bitsleft = 8, byte = random_byte();
        v = byte & 1;
        byte >>= 1;
        bitsleft--;
        bignum_set_bit(w, i, v);
      }
      bn_restore_invariant(w);
      if (bignum_cmp(w, p) >= 0 || bignum_cmp(w, Zero) == 0) {
        freebn(w);
        continue;
      }
      break;
    }

    pfn(pfnparam, PROGFN_PROGRESS, phase, ++progress);

    /*
     * Compute w^q mod p.
     */
    wqp = modpow(w, q, p);
    freebn(w);

    /*
     * See if this is 1, or if it is -1, or if it becomes -1
     * when squared at most k-1 times.
     */
    if (bignum_cmp(wqp, One) == 0 || bignum_cmp(wqp, pm1) == 0) {
      freebn(wqp);
      continue;
    }
    for (i = 0; i < k - 1; i++) {
      wqp2 = modmul(wqp, wqp, p);
      freebn(wqp);
      wqp = wqp2;
      if (bignum_cmp(wqp, pm1) == 0)
        break;
    }
    if (i < k - 1) {
      freebn(wqp);
      continue;
    }

    /*
     * It didn't. Therefore, w is a witness for the
     * compositeness of p.
     */
    freebn(wqp);
    freebn(p);
    freebn(pm1);
    freebn(q);
    goto STARTOVER;
  }

  /*
   * We have a prime!
   */
  freebn(q);
  freebn(pm1);
  return p;
}

/*
 * Invent a pair of values suitable for use as 'firstbits' in the
 * above function, such that their product is at least 2.
 *
 * This is used for generating both RSA and DSA keys which have
 * exactly the specified number of bits rather than one fewer - if you
 * generate an a-bit and a b-bit number completely at random and
 * multiply them together, you could end up with either an (ab-1)-bit
 * number or an (ab)-bit number. The former happens log(2)*2-1 of the
 * time (about 39%) and, though actually harmless, every time it
 * occurs it has a non-zero probability of sparking a user email along
 * the lines of 'Hey, I asked PuTTYgen for a 2048-bit key and I only
 * got 2047 bits! Bug!'
 */
void invent_firstbits(unsigned *one, unsigned *two)
{
  /*
   * Our criterion is that any number in the range [one,one+1)
   * multiplied by any number in the range [two,two+1) should have
   * the highest bit set. It should be clear that we can trivially
   * test this by multiplying the smallest values in each interval,
   * i.e. the ones we actually invented.
   */
  do {
    *one = 0x100 | random_byte();
    *two = 0x100 | random_byte();
  } while (*one * *two < 0x20000);
}
